Formal Properties of Metrical Structure 
Marc van Oostendorp 
Werkverband Grammaticamodellen 
Tilburg University 
P.O.Box 90153 
5000 LE Tilburg 
The Netherlands 
oostendo~kub.nl 
Abstract 
This paper offers a provisional mathemat- 
ical typology of metrical representations. 
First, a family of algebras corresponding 
to different versions of grid and bracketed 
grid theory is introduced. It is subsequently 
shown in what way bracketed grid theory 
differs from metrical theories using trees. 
Finally, we show that there are no sig- 
nificant differences between the formalism 
of bracketed grids (for metrical structure) 
and the representation used in the work of 
\[Kaye, et al., 1985\], \[1990\] for subsyllabic 
structure. 
1 Introduction 
The most well-known characteristic of Non-linear 
Phonology is that it shifted its attention from the 
theory of rules (like in \[Chomsky and Halle, 1968\]) to 
the theory of representations. During the last decade 
phonologists have developed a theory of representa- 
tions that is sufficiently rich and adequate to describe 
a wide range of facts from the phonologies of various 
languages. 
It is a fairly recent development that these repre- 
sentations are being studied also from a purely for- 
mal point of view. There has been done some work 
on autosegmental structure (for instance \[Coleman 
and Local, 1991; Bird, 1990; Bird and Klein, 1990\]) 
and also some work on metrical trees (like \[Coleman, 
1990; Coleman, 1992\] in unification phonology and 
\[Wheeler, 1981; Moortgat and Morrill, 1991\] in cat- 
egorial logics). As far as I know, apart from the pio- 
neering work by \[Halle and Vergnaud, 1987\], hitherto 
no attention has been paid to the formal aspects of 
the most popular framework of metrical phonolog~y 
nowadays, the bracketed grids framework. 
Yet a lot of questions have to be answered with 
regard to bracketed grids. First of all, some authors 
(for instance \[Van der Hulst, 1991\]) have expressed 
the intuition that bracketed grids and tree structures 
(e.g. the \[sw\] labeled trees of \[Hayes, 1981\] and re- 
lated work) are equivalent. In this paper, I study 
this intuition in some formal detail and show that it 
is wrong. 
Secondly, one can wonder what the exact rela- 
tion is between higher-order metrical structure (foot, 
word) and subsyllabic structure. In this paper I will 
show that apart from a fewempirically unimportant 
details, bracketed grids are equivalent to the kind of 
subsyllabic structure that is advocated by \[Kaye, et 
al., 1985\], \[1990\] t. 
2 The definition of a bracketed grid 
Below I give a formal definition of the bracketed grid, 
as it is introduced by HV and subsequently elabo- 
rated and revised by these authors and others, most 
notably \[Hayes, 1991\]. HV have a major part of their 
book devoted to the formalism themselves, but there 
are numerous problems with this formalization. I 
will mention two of them. 
First, their formalization is not flexible enough to 
capture all instances of (bracketed) grid theory as 
it is actually used in the literature of the last few 
years. They merely give a sketch of the specific im- 
plementation of bracketed grid theory as it is used 
in the rest of their book. Modern work like \[Kager, 
1989\] or \[Hayes, 1991\] cannot be described within 
1In this paper, I will use HV as an abbreviation for 
\[Halle and Vergnaud, 1987\] and KLV as an abbreviation 
for \[Kaye, et al., 19851, \[1990\]. 
322 
this framework. 
Secondly, their way of formalizing bracketing grids 
has very much a 'derivational' flavour. They are 
more interested in how grids can be built than in 
what they look like. Although looking at the deriva- 
tional aspects is an interesting and worthwile enter- 
prise in itself, it makes their formalism less suitable 
for a comparison with metrical trees. 
A grid in the linguistic literature is a set of lines, 
each line defining a certain subgroup of the stress 
bearing elements. Thus, in 1 (HV's (85)), the as- 
terisks ('stars') on line 0 represent the syllables of 
the word formaldehyde, the stars on line 1 secondary 
stress and line 2 represents the syllable with primary 
stress: 
(1) (. *) line 2 
(*) (* .) (*) line 1 
for mal de hyde 
We can formalize the underlying notion of a line as 
follows: 
Definition 1 (Line) A line Liis a pair < At, -'4i> 
where Ai = {a~,...,a?}, where a~,...,a n are con- 
stants, n a fixed number 
-~i is a total ordering on Ai such that the following 
axioms hold. 
a. Vot, 13, 7 ELi : ot ~i 13 A 13 ~i 7 ~" ot -41 7 (transi- 
tivity) 
b. Vow, 13 ELi : a -4i 13 ::~ -'(13 "~i a) (asymmetry) 
c. Va G Li : ~(a -41 a) Orreflezivity) 
We say that Li C Lj if Ai C A#, a ELi if a E At. 
Other set theoretic expressions are extended in a like- 
wise fashion. 
Yet this formalisation is not complete for bracketed 
grids. It has to be supplemented by a theory about 
the brackets that appear on each line, i.e. by a the- 
ory of constituency and by a theory of what exactly 
counts as a star on a given line. 
We have exactly one dot on top of each column of 
stars2.Moreover, each constituent on a line has one 
star in it plus zero, one or more dots. The stars 
are heads. HV say that these heads govern their 
complements. This government relation can only be 
a relation which is defined in terms of precedence. 
Suppose we make this government relation into the 
primitive notion instead of the constituent. A metri- 
cal line is defined as a line plus a government relation 
on that line: 
Definition 2 (Metrical line) A metrical line 
MLi is a pair <Li,P~>, where 
Li is a line 
Ri is a relation on Li and an element of {-~i,>-i,--,i 
, ~-'i, '~i} 
ZThat is, if we follow the current tradition rather than 
HV. 
With the following definitions holding3: 
Definition 3 (Precedence Relations) a Ni 13 ¢~ 
We assume that something like Government Require- 
ment 1 holds, just as it is assumed in HV that every 
element is in a constituent, modulo extrametricality 
(which we will ignore here). 
Government Requirement 1 (to be revised be- 
low) A line Limeets the government requirement iff 
all dots on Li are governed, i.e. a star is in relation 
Ri to them. 
Now a constituent can be defined as the domain that 
includes a star, plus all the dots that are governed 
by this star. We have to be a little bit careful here, 
because we want to make sure that there is only one 
star in each constituent. 
In a structure like the following we do not want 
to say that the appointed dot is governed by the 
first star. It is governed by the second star, which is 
nearer to it: 
(2) *...*... 
T 
In order to ensure this, we adopt an idea from mod- 
ern GB syntax, viz. Minimality, which informally 
says that an element is only governed by another el- 
ement if there is no closer governor. The definition 
of Phonological Minimality could look as follows: 
Definition 4 (phonological government) aGi# 
(a governs 13 on line i) iff a is a star and 
art13 A -~37, 7 a star : \[7R/13 A aR/7\] 
We will give the formal definition of a star later on 
in this chapter. The government requirement is now 
to be slightly modified. 
Government Requirement 2 to be revised below 
A line Limeets the government requirement iff all 
dots on Li are governed, i.e. a star in Liis in relation 
Gi to them. 
We can now formally define the notion of a 
constituent 4. 
aActually, HV also use a fifth kind of constituent in 
their book, viz. one of the form (. * .). Because there 
has been a lot of criticism in the literature against this 
type of government, I will not not discuss it here. 
4The reviewer of the abstract for EACL notices that 
under the present definitions it is not possible to ex- 
press the kind of ambiguity that is current in (parts of) 
bracketed grid literature, where it is not sharply defined 
whether a dot is governed by the star to its left or by the 
star to its right. This is correct. It is my present purpose 
to define a version of bracketed grids that comes closest 
to trees because only in this way we can see which are the 
really essential differences between the two formalisms. 
323 
Definition 5 (Constituent) A constituent on a 
line Li is a set, consisting of exactly one star S in 
Li plus all elements that are not stars but that are 
governed by S. 
We now have a satisfying definition of a metrical 
line. We can define a grid as a collection of metrical 
lines, plus an ordering relation on them: 
Definition 6 (Grid) A grid G is a pair < £, 1>, 
where 
£ = {L1 .... , Ln}, where LI, ..., L, are metrical lines 
I is a total ordering on £, such that VLi, Lj • £ : 
Li I L1 ¢~ \[Li C Lj A Va,~3 • Li fl Lj : \[a -~i Z ¢~ 
8\]\] where C is intended to denote the proper subset re- 
lation, so Li C L 1 ~ ",(Li = Li). 
It is relatively easy to see that I by this definition is 
transitive, asymmetric and irreflexive. We also define 
the inverse operator T such that Li T L 1 iff Lj ILi. 
The most interesting part of definition 6 is of 
course the 1 ('above')-relation. Look at the grid in 
(3) (= HV's (77), p. 262): 
(3) line 3 
(. .) line 2 
(. .) (.) line 1 
Ten ne see 
Each of the lines in this grid is shorter than the one 
immediately below it, in that it has fewer elements. 
This follows from elementary pretheoretical reason- 
ing. Every stressed syllable is a syllable, every syl- 
lable with primary stress also has secondary stress. 
We expressed this in definition 6 by stating that ev- 
ery line is a subset of the lines below it. By this 
statement we also expressed the idea that the ele- 
ments represented on the higher line are in fact the 
same things as those represented on the lower lines, 
not just features connected of these. The second part 
of the definition says that the relative ordering of the 
elements in each line is the same as that on the other 
lines. 
Our present definition of a metrical grid already 
has some nice properties. For example, the Continu- 
ous Column Constraint, which plays a crucial role as 
an independent stipulation in \[Hayes, 1991\] can be 
derived from definition 6 as a theorem: 
(4) Continuous Column Constraint (CCC): A 
grid containing a column with a mark on 
layer (=our metrical line) n+l and no mark 
on layer n is ill-formed. Phonological rules 
are blocked when they would create such a 
configuration. 
The CCC excludes grids like (5), where b is present 
on the third line, but not on the second. 
(5) a b 
a c d 
abed 
We can formalize the CCC as a theorem in our sys- 
tem: 
Theorem 1 (CCC)VaVLiLj : (a E Li ALi \]. 
e 
Proof of theorem 1 Suppose a E Li, suppose 
Li I Lj. Then (by (13)) L, C Lj. Now the stan- 
dard definition of C implies VX : X ELi -"* X E Lj. 
Instantiation of X by c~, our first assumption and 
Modus Ponens give a E Li.O 
We can also easily define the notion of a dot and 
a star, informally used in the above definitions of 
government. 
Definition 7 (Star and dot) 1. Va E Li : 
stari(ot)de--~f3Lj : ILl ~ Lj A (a E Lj)\] 
def e. Va ELi : doti(ot)= ~stari(ot) 
Government Requirement 2 can now be fully for- 
malised and subsequently extended to the grid as 
a whole. 
Government Requirement 3 (for lines) -- fi- 
nal version A metrical line Li meets the government 
requirement iff Va ELi : doti(a) =~ 38 E Li : 
^ 
Government Requirement 4 (for grids) -- to 
be revised below A grid G meets the government re- 
quirement iff all lines in G meet the government re- 
quirement. 
We want to introduce an extra requirement on grids. 
Nothing in our present definition excludes grids con- 
sisting of infinitely many lines. However, in our lin- 
guistic analyses we only consider finite construetions. 
We need to express this. First, we define the notions 
of a top line and a bottom line. Then we say that a 
finite grid always has one of each. 
Definition 8 (Top line and bottom line) For a 
certain grid G, VLi E G 
(LToP, G = Li)d--efVLj E G : \[(Li = Lj) V (Li ~ Lj)\] 
(LBoTTOM,G -" LI)~'~fVLj • G : \[(ni = Lj)V(Lj 1 
Li)\] 
Definition 9 (Finite grid) A grid G is called a fi- 
nite grid if 
3Li • G: \[Li= LTOP, G\] A 3Lj • G : \[L i = 
LBoTTOM,e\] 
Note that we have to say something special with re- 
gard to the government relation in LTOP, G. By defi- 
nition, this line has only dots in it, so it always looks 
as something like (6). 
(6) ....... 
There can be no star on this level. A star by defini- 
tion has to be present at some higher line and there 
is no higher line above LTOP, G. This means that 
the LTOP, G can never be meeting the government 
324 
requirement and that in turn means that no linguis- 
tic grid can ever meet the government requirement. 
In order to avoid this rather unfortunate situation, 
we have to slightly revise the definition of meeting 
the government requirement for linguistic grids. 
Government P~quirement 5 (for grids) -- fi- 
nal version A grid G meets the government relation 
iff all the lines Li E G - {LTop, a} meet the govern- 
ment requirement. 
Definition 10 (Linguistic grid) A linguistic grid 
is a finite grid which meets the government relation 
A last definition may be needed here. If we look at 
the grids that axe actually used in linguistic theory, 
it seems that there is always one line in which there 
is just one element. Furthermore, this line is the top 
line (the only line that could be above it would be an 
empty line, but that one doesn't seem to have any 
linguistic significance). 
This observation is phrased in \[Hayes, 1991\] as fol- 
lows: if prominence relations are obligatorily defined 
on all levels, then no matter how many grid levels 
there are, there will be a topmost level with just one 
grid mark. 
We can formalize this ~s follows: 
Definition 11 (Complete linguistic grids) 
A linguistic grid G is called a complete linguistic 
grid iff \[LToP, G\] = 1, i.e. 3a : \[a E LTOP, G A Vfl : 
L 8 e LTOP, G ::~ # ---- O~\]\] 
We call this type of grid complete because we can eas- 
ily construct a complete linguistic grid out of every 
linguistic grid. 
If LTOP, G is non-empty, we construct a complete 
grid by projecting the rightmost (or alternatively the 
leftmost) element to a new line L/and by adding the 
government requirement ~- (or -4) to LTOP, G. Fi- 
nally we add the relation Lil LTOP, G to the grid, 
i.e. we make Lito the new LTOP, G. 
If the top line of the grid is empty, we remove 
this line from the grid and proceed as above. Most 
linguistic grids that are known from the literature, 
are complete. 
Some authors impose even more restrictions on 
their grids. I believe most of those claims can be 
expressed in the formal language developed in this 
section. One example is \[Kager, 1989\], who claims 
that all phonological constituents are binary. This 
Binary Constituency Hypothesis can be formulated 
by replacing definition 2: 
Definition 12 A metrical line MLI is a pair < 
Li,Ri>, where 
Li is a line 
Ri is a relation on Li and an element of {~i, ~i} 
3 Grids and trees 
In this section, we will try to see in how much brack- 
eted grids and trees are really different formal sys- 
terns, i.e. to what extent one can say things in one 
formalism that are impossible to state in the other. 
First recall the standard definition of a tree (we 
cite from \[Partee et al., 1990\])5: 
Definition 13 (Tree) A 
(constituent structure) tree is a mathematical con- 
figuration < N, Q, D, P, L >, where 
N is a finite set, the set of nodes 
Q is a finite set, the set of labels 
D is a weak partial order in N × N, the dominance 
relation 
P is a strict partial order in N x N, the precedence 
relation 
L is a function from N into Q, the labeling function 
and such that the following conditions hold: 
(a) 3a E N : V/~ G N : \[<a,/~>E O\] (Single root 
condition) 
(b) W,a ~ N : \[(<~,~>~ PV <a,~>E P) ¢* (< 
a,/~ >¢ D^ </~, a >¢ D)\] (Exclusivity condi- 
tion) 
(c) Va,~,7,6 : \[(< ot,/~ >E PA < a,7 >E DA < 
8, 6 >E D) ~< 7, 6 >E P\] (Nontangling condi- 
tion) 
It is clear that bracketed grids and trees have 
structures which cannot be compared immediately. 
Bracketed grids are pairs consisting of a set of com- 
plex objects (the lines) and one total ordering rela- 
tion defined on those objects (the above relation). 
Trees on the other hand are sets of simple objects 
(the nodes) with two relations defined on them (dom- 
inance and precedence). These simply appear to be 
two different algebra's where no isomorphism can be 
defined. 
Yet if we decompose the algebraic structure of the 
lines, we see that there we have sets of simple objects 
(the elements of the line) plus two relations defined 
on them. One of those relations ('~i) is a strict par- 
tial order, just like P. The other relation, Gi, vaguely 
reminds us of dominance. 
Yet a line clearly is not a tree. Although -4i has 
the right properties, it is not so sure that Gi does. 
While this relation clearly is asymmetric (because it 
is directional), it is not a partial order. 
First of all, it is not transitive. (7) is a counterex- 
ample. 
(7) • • +-- 
a b c 
Here aGib and bGie but not aGie, because of min- 
imality (there is a closer governor, viz. b). Gi also 
5For the moment, we will not consider Q and L, be- 
cause these are relatively unimportant for our present 
aim and goal and there is nothing comparable to the la- 
beling function in our definition of bracketed grids. This 
is to say that for now we will study unlabeled trees. Notice 
however that the trees actually used in the phonological 
literature do use st least a binary set of labels { s, w } 
325 
is irreflexive, of course, because no element is to the 
left or to the right of itself. 
A more interesting relation emerges if we consider 
the grid as a whole. Because trees are finite struc- 
tures, we need to consider linguistic grids only. The 
line LBOTTOM,G has the property that VaVLI ~ G : 
\[a ~Li =~ ~ ~ LBOTTOM,G\]. This follows from the 
definitions of LBOTTOM,G and of the 'above' rela- 
tion. 
This means that all basic elements of the grid are 
present on LBOTTOM,G and, as we have seen above, 
we can equal P to "~BOTTOM,G. Furthermore, we 
can build up a 'supergovernment' relation {7, which 
we define as the disjunction of all government rela- 
tions Gi in G. 
Definition 14 (Supergovernment) 
{70 d¢__f U {<a,f~> laG~3Adot~(~)} 
LiEG 
Again, we exclude the government relation of the two 
stars in (7). 
If we want to compare {7 to dominance, we have to 
make sure it is a partial order. However, {7 obviously 
still is irreflexive. It also is intransitive. Consider the 
following grid for example. 
(8) • 
a cd 
In this grid a{Tc A c{Td but --,a{Td. For this reason, we 
take the transitive and reflexive closure of {7, which 
we call 7"7¢{7. 
LFrom this, we can define the superline of a lin- 
guistic grid 6. 
Definition 15 (Superline) The superline S/~ of a 
linguistic grid G is the tuple 
< ABOTTOM,G, "~BOTTOM,G, "Jf'T~{TG >. 
The superline is an entity which we can for- 
mally compare to a tree, with -~BOTTOM,G = P, 
ABOTTOM,G = N, 7"T~{7 = D. Of most interest are 
the complete linguistic grids, firstly because these are 
the ones that seem to have most applications in lin- 
guistc theory and secondly because the requirement 
that they be complete (i.e. their LTOP, G should have 
exactly one element) mirrors the single root condi- 
tion on trees. From now on, we will use the abbrevi- 
ation CLG for 'complete linguistic grid'. 
Note that we also restrict our attention to grids 
which meet the government requirement, i,e. to lin- 
guistic grids. We are not so sure that this restriction 
is equally well supported by metrical theory as the 
restriction to completeness. However, the restriction 
6The superfine itself has no specific status in linguistic 
theory. I also do not claim it should have one. The 
superfine is a formal object we construct here because 
it is the substructure of the bracketed grid that comes 
closest to a tree. 
to linguistic grids makes sure that all elements in the 
grid participate in the government relation, because 
everything ends as a star somewhere and hence has 
to be governed by another element. 
In order to somewhat simplify our proofs below, we 
introduce one new notational symbol here: ~. 
Definition l6 (Top Line)V~ E AVLi ~ G : 
\[c~Li ¢~ c~ ~ Li A -,~Lj \[a ~ Lj A L~ ~ Li\]\] 
This symbol '_~' 'top line' denotes the highest line on 
which a certain element is present. If a ~ Li, then 
Liis the highest line at which a can be found. By 
definition, this means that ~ is a dot on Li. 
Of course, for every element in a linguistic grid 
there is one specific top line. 
We now prove: 
Theorem 2 For every linguistic grid G, ifG is com- 
plete, then SLG satisfies the Single Root Condition. 
Proof of theorem 2: Consider a complete linguis- 
tic grid G. We have to prove that 3a E A : V~ E 
A : \[< a,/3 >E "/-T~{TG\] (for shortness, we will refer 
here and in the following to ABOTTOM,G as A and to 
"~BOTTOM,G as "~ where no confusion arises). Con- 
sider the (single) element of LTOP.a. We call this el- 
ement 7 and prove that V/~ E A : I<7,/~>E TT~gG\]. 
(Reductio ad absurdum.) Suppose 3/~ E A : \[< 
7,/~>~ TT~{Ta\]. Because this/~ is in A, 3Li : ~_ELi\]. 
We now take the highest/3 for which this condition 
is true, i.e. 
V~ E A : \[<7,~>~ TT~{TG =~ 3Lk : \[~E_L~ALk T Li\]\] 
/~_ELI by definition means that there is no Lj higher 
than Li of which/~ is a member. But this in turn 
means that/3 is a dot on Li (or doti(/~)). 
Li cannot be equal to LTOP, G, because in that case 
we would have/3 -- 7 and since 7"~{TGis reflexive, 
< 7,/3 >E TT~{TG, contrary to our assumption. So 
LTOP, G ~ Li 
Because doti(/~) and the grid meets the govern- 
ment requirement 36 : \[stari A 6Gi7\]. From the defi- 
nition of supergovernment we then get that < 6,/~>E 
T ~{T G 
6 is a star on Li. This means that 3Lm : \[6~Lm A 
Lm ~ Li\]. We can conclude now that < 7,6 >E 
TT~{TG holds, because delta is on a higher line than 
/3 and we assumed /3 was the highest element for 
which this condition did not hold. 
But now we have < 6, fl >E q'T~{TaA < 3', 6 >E 
TT~{TG and because TT~{Tais transitive, < %/3 >E 
"/'7~{7G. This is a contradiction with our initial as- 
sumption. \[\] 
So superlines have one important characteristic of 
trees. Yet exclusivity and nontangling still do not 
hold for superlines of CLGs, even if they meet the 
government requirement. 
A counter example to exclusivity is (9), where a -~ 
bA <a,b>E TT~{TG. 
326 
(9) • *-- 
a b 
A counterexample to the nontangling condition is 
(10), where a -~ bA < a,c >• 7"TCGcA < b,c >• 
q'7~Ga but c ~ c. 
(10) * ~- 
• * +-- 
a b c 
The reason why these conditions do not hold is that, 
on lines as well as on superlines, elements can both 
govern and precede another element. Exclusivity and 
nontangling are meant to keep precedence and dom- 
ination apart. 
Sometimes in the literature on trees (e.g. Samp- 
son 1975) we find some weakening of the definition 
of a tree, in which exclusivity and nontangling are 
replaced by the Single mother condition. 
We first define the mother relation, which is im- 
mediate dominance: 
Definition 17 (Mother) -- to be revised below 
For all T, T a tree 
Va/9 • T\[aM/9 ¢~< a,/9 >• D\[= TT~G\] A -~37 : \[< 
a,7>• D\[= 7"T~6G\]A <%/9>6 D\[= 7-TdgG\]\]\] 
Definition 18 (Single mother condition) 
Va,/9, 7:\[(aM~9 A 7M/9) ¢=> (a = 3')1 
Because 7-~G is the transitive closure of TONG, we 
can rephrase defintion 17) as definition 19 for super- 
lines of CLGs: 
Definition 19 (Mother) -- final version For all 
snperlines 8 f.G 
Va/9 6_ S£a\[aMt~ ¢~< a,/9>6 7~#a\] 
We can now prove: 
Theorem 3 For every grid G, if G is a CLG then 
Sf.G satisfies the Single Mother Condition. 
Proof of theorem 3: (By RAA.) Suppose a,/9, 7 • 
S£G and aM/9 A 7M/9 A a # 7- If aM/9, then by 
definition 19, < a,/9>• ~Ta, and if 7M/9, similarily 
<7,/9>6 TONG. Because a # 7, we have a # /9. 
For if a =/9, we would have 7Ma A aM~9. But by 
(19) we then cannot have 7M/9. A similar line of 
reasoning shows that/9 ~ 7. So a ¢ 7 A 7 ¢/9. By 
(17) this means that \[< a,/9>6 ~GA < 7, ~ >6 Ca\], 
because 7ZQG is the transitive closure of GG. We 
have reached the following proposition: 
Proposition 1 The mother relation equals ~ on su- 
perlines: V~/9 • SfG : \[otMfl =~< a,/~>6 ~G\] 
Definition 14 says that if < a,/9>• fig there is a line 
L/such that aGi/gAdoti(~)}. Also, if < %/9 >• Ga 
there is a line Ljsuch that 7Gj/gAdob(#)}. Because 
can by definition be a dot at exactly one line, Li= 
Ljand otGi/9 and 7Gi/9. However, from the minimal- 
ity definition of government (4), it follows that in 
that case (~ = 7. Which is a contradiction. \[\] 
4 Dependency Trees 
Let us summarize the results so far. We have seen 
that from bracketed grids we can extract sup•tithes, 
on which the government relations of the normal lines 
are conflated. 
These superlines are equivalent to some sort of un- 
labeled trees, under a very weak definition of the lat- 
ter notion. Whereas the minimal restrictions of the 
Single Root Condition and the Single Mother Condi- 
tion do hold, the same is not necessarily true for the 
Exclusivity Condition and the Non-Tangling Condi- 
tion. 
It can be shown that in the linguistic literature 
a form of tree occurs that is exactly isomorphic to 
bracketed grids. These are the trees that are used in 
Dependency Phonology. 
We did not yet discuss what the properties of these 
trees are. This is what we will briefly do in the 
present section. 
First let us take a look at the kind of tree we can 
construct from a given grid. We give the CLG in 
(11) as an example: 
(11) LTOP, G 
* b- 
e? 
/,From this grid we can derive a superline 8£a 
with {~G = {<a,b>,<c,d>,<e,f>,<e,a>,< 
e, c >}. If we interpret this as a dominance relation 
and if we draw dominance in the usual way, with the 
dominating element above the dominated one, we get 
the following tree: 
(12) e 
b d f 
This tree looks rather different than the structures 
used in the syntactic literature or in metricM work like \[Hayes, 1981\]. 
Yet there is one type of structure known in the 
linguistic (phonological) literature which graphically 
strongly resembles (12). These are the Defendency Graphs 
(DGs) of Dependency Phonology (\[Durand, 
1986\] a.o.). 
According to \[Anderson and Durand, 1986\], DGs 
have the following structure. They consist of a set 
of primitive objects together with two relations, de- 
pendency and precedence. For example within the 
syllable/set/the following relations are holding (no- 
tice some of the symbols we introduced above are 
used here with a slightly different interpretation): 
Dependency s ,--- e --* t (i.e. /s/ depends on/e/ 
and/t/depends on ~el. 
327 
Precedence s < e < t (i.e. /s/ bears a relation 
of 'immediate strict precedence to/e/which, in 
turn, bears the same relation to/t/. 
Anderson and Durand also introduce the transi- 
tive closure of 'immediate strict precedence', 'strict 
precedence', for which they use the symbol << and 
the transitive closure of dependency, 'subordination', 
for which they use the double-headed arrow. More- 
over, well-formed dependency graphs conform to the 
following informal characterisation (=Anderson and 
Durand's (10)): 
Definition 20 (Dependency graph) 
(---Anderson and Durand's (10)) 
1. There is a unique vertex or root 
2. All other vertices are subordinate to the root 
3. All other vertices terminate only one arc 
4. No element can be the head of two different con- 
structions 
5. No tangling of arcs or association lines is al- 
lowed 
20.1 and 20.2 together form a redefinition of the 
Single Root Condition. Theorem 2 states that su- 
perlines of CLGs with the government requirement 
satisfy this Condition. 
Because 'arcs' are used as graphic representations 
for dependency (which is intransitive), 20.3 seems a 
formulation of the Single Mother Condition. Theo- 
rem 3 states that this condition also holds for super- 
lines of complete linguistic grids. 
20.4 needs some further discussion because it is 
the only requirement that does not seem to hold for 
our grids. The condition says that something cannot 
be a head at more than one level of representation, 
e.g. something cannot be the head of a foot and of 
a word. However, because of the CCC, in bracketed 
grid systems the head of a word always is present (as 
a star - hence as a head) on the foot level. 
It is exactly this requirement that is abandoned by 
all authors of at least Dependency Phonology. \[An- 
derson and Durand, 1986\] (p.14) state that one el- 
ement can be the head of different constructions, as 
indeed we have already argued in presenting a given 
syllabic as suecesively the head of a syllable, a foot 
and a tone group. 
In order to represent this, a new type of relation 
is introduced in their system, subjunction. A node a 
is subjoined to/3 iff (~ is dependent on/3 but there is 
no precedence relation between the two. 
The word intercede then gets the following repre- 
sentation: 
(13) o 
O O b,, 
o o I I I 
in ter cede 
We once again cite \[Anderson and Durand, 1986\] (p.15): 
The node of dependency degree 0 is ungoverned 
(the group head). On the next level down, at de- 
pendency degree 1, we have two nodes governed by 
the DDO node representing respectively the first foot 
(inter) and the second foot (cede). The first node 
is adjoined to the DDO node, the second one is sub- 
joined. Finally, on the bottom level, at DD2, the 
nodes represent the three syllables of which this word 
is comprised. These latter nodes are in turn governed 
by the nodes at DD1 and once again related to them 
by either adjunction or snbjunction. 
Lifting the restriction this way seems to be exactly 
what is needed to fit the superline into the DG for- 
malism. 
20.5 holds trivially in the bracketed grid frame- 
work as well. It can be interpreted as: if ~ precedes 
/3 on a given line in the grid, there is no other line 
such that/3 precedes c~ on that line. This is included 
in the definition of the '~' relation, 
The difference between a phonological DG and 
a bracketed grid is the same as the difference be- 
tween a superline and a bracketed grid: the DG is 
not formally divided into separate lines. Interest- 
ingly, \[Ewen, 1986\] analyses English stress shift, one 
of the main empirical motivations behind the grid 
formalism, with subjunction. In \[Van Oostendorp, 
1992b\] I argue that using subjunction Ewen's way 
actually means an introduction of lines into Depen- 
dency Phonology. 
5 Government Phonology 
Now let us turn over to a well-known phonological 
theory that also employs the notion of government 
as well as 'autosegmental representations'. I refer of 
course to the syllable theory of KLV and \[Charette, 
1991\]. 
This theory of the syllable in fact does not have 
a syllable constituent at all. In stead of such a con- 
stituent, KLV postulate a line of x-slots and par- 
alelly, a tier of representation which conforms to the 
pattern (0 R)* - that is an arbitrary number of rep- 
etitions of the pattern OR.(KLV \[1990\]) The term 
'tier' suggests an autosegmental rather than a met- 
rical (bracketed grid) approach to syllable structure, 
but KLV are never explicit on this point. 
The fact that O and R appear in a strictly regular 
pattern can be explained either by invoking the (met- 
rical) Perfect Grid requirement or, alternatively, the 
328 
(autosegmental) OCP. The same applies to the 'la- 
bels' O and R: we can define them autosegmentally 
as the two values of a type 'syllabic constituent' or in- 
directly as notational conventions for stars and dots 
on a 'syllable line', i.e. we could have the following 
representations for KLV's (O R)* line: 
(14) a. tier:. 
\[type: syll.const; vMue: O;...\] \[type: syll. coast; 
v~lue: R;... \], etc. 
b. line: 
*, etc. 
For (14b), we would have to show that the rhymes 
or nuclei project to some higher line. We will return 
to this below. 
We still cannot really decide between an autoseg- 
mental and a metrical approach. If we look at more 
than one single line, this situation changes. 
At first sight, it then seems very clear that KLV's 
syllables act as ARs, not as grids. For instance, we 
can have representations like (15) (from \[Charette, 
1991\]), with a floating Onset constituent: 
(15) Ort O R 
I I I 
X X X 
I I I a m 1 
This is a possible autosegmental chart, but not 
a possible grid (because the Complex Column Con- 
straint is violated by the word initial onset). How- 
ever, empirical motivation for (15) is hard to find. As 
far as I know, the structure of (15) is motivated only 
by the assumption that on the syllabic line we should 
find (OR)* sequences rather than, say, (R)(OR)*. 
The same state of poor motivation does not hold, 
however, to the representation \[Charette, 1991\] as- 
signs to words with an 'h aspir6'7: 
(16) O R 
I I 
X X X 
I I a § 
As is well known, the two types of words behave 
very differently, for example with regard to the def- 
inite article. While words with a lexical represen- 
tation as in (16) behave like words starting with a 
'real', overt, onset, words with a representation like 
(15) behave markedly different: 
(17) a. le tapis- *l' tapis 
b. la hache - *l' hache 
c. *la amie - l' amie 
It seems that, while the empty onset of (15) is 
invisible for all phonological processes, the same is 
not true for the empty onset of (16). 
rI disregard the (irrelevant) syllabic status of the final 
\[§\] consonant. 
So there are two different 'empty onsets' in KLV's 
theory s. Notice that the type of empty onset for 
which there is some empirical evidence is exactly the 
one where the (0 R) - x slot chart does behave like 
a grid (i.e. where it does not violate the CCC). 
So whereas we have here a formal difference be- 
tween KLV's theory and grid theory, this has no real 
empirical repercussions. 
Another similarity is of course the notion 'govern- 
ment'. For KLV, government only plays a role on 
the line of x-slots. \[Charette, 1991J(p. 27) gives the 
following summary: 
Governing relations must have the following prop- 
erties: 
(i) Constituent government: the head is initial and 
government is strictly local. 
(it) Interconstituent government: the head is final 
and government is strictly local. 
Government is subject to the following properties: 
(i) Only the head of a constituent may govern. 
(il) Only the nuclear head may govern a constituent 
head. 
The most important government relation is con- 
stituent government: this is the relation that defines 
the phonological constituent. Moreover, the 'prin- 
ciples' given by Charette are only introduced into 
the theory to constrain interconstituent government. 
By definition, constituent government remains unaf- 
fected by these. (As for (i), the definition of the no- 
tion constituent implies that it is only the head that 
governs and (it) does not apply because we never find 
two constituent heads within one constituent). 
The two conditions on constituent government 
(that the head be initial and the governee adjacent 
to it) can be expressed in our formalisation of the 
grid in a very simple way: 
Definition 21 Rx-stot =~'-" 
According to KLV, *-- is the only possible con- 
stituent government relation. Other candidates like 
{---~, ~-,-~,,~} are explicitly rejected, so in fact we 
have (with some redundancy): 
Definition 22 VLi : \[R/E {~-)\] A R~-,lot =*'-- 
8\[Piggot and Singh, 1985\] propose a different distinc- 
tion, namely one in which the empty onset of ami is rep- 
resented as (in) and the one of hache as (ib) (0 is a null 
segment): 
(i) a. 0 b. 0 
I I 
X X 
I 
0 
Under this interpretation of Government Phonology, 
the syllable structure is formally even more similar to 
grids, if we assume that the linking between segmental 
material and x-slots has to be outside the grid (treated 
as autosegmental association) anyway. 
329 
This is one of the reasons why KLV do not accept 
the syllable as a constituent: under their definition 
of government, this would make the onset into the 
head of the syllabic constituent. 
At least we can see from these definitions that the 
x-slot line in KLV's theory behaves like a normal 
metrical line. 
Yet there is one extra condition defined on this 
line; this is called interconstituent government. Be- 
cause of the restrictions in (15), KLV notice that this 
type of government only concerns the following con- 
texts. (Square brackets denote domains for intercon- 
stituent government, normal brackets for constituent 
government): 
(18) a. N O A I 
(x Ix) (x\] 
b. NON 
I I \[(x) (xl 
c. 0 R 
I I Ix) (x\] 
But the fact that there is an extra condition on a 
line does not alter its being metrical, even if we call 
this extra condition a government relation 9. 
We now have reached the following representation 
(20) of the grid variant of the x line in (19) (we use 
the star-and-dot notation and leave out the associa- 
tion of the autosegmental material to the skeleton): 
(19) O R O R 
I IAi 
X X X X X 
I I I I I p a t r l 
(20) O R O tt (*) (*) (* .) (*) 
By definition, stars are present on a higher line. 
As we have seen above, there is no reason not to 
consider the (O R)* tier to be this higher line. We 
then get the following representation: 
(21) (. *) (. *) line 2 (?) 
(*) (*) (* .) (*) line l(~) 
p a t r i 
As we noted above, KLV do not accept any con- 
stituents on the higher line. One of their reasons was 
their stipulation that all constituents are left-headed. 
There are independent reasons to abandon this re- 
striction. \[Charette, 1991\] argues for a prosodic anal- 
ysis of French schwa/\[e\] alternations. In order to do 
this, she has to build metrical (Is w\] labeled) trees 
representing feet on top of the nuclei. She gives the 
9In \[Van Oostendorp, 1992a\] I sketch a way of trans- 
lating 'interconstituent government' to a bracketed grid 
theory of the syllable. 
following crucial example \[Charette, 1991\] (p.180, ex- 
ample (llc)): 
(22) F 
W S 
I I 
N N 
I I 0 N O N 
I I I I 
X X X X 
I I I I 
m ~ n e 
Here we have a clear case of a right-headed phono- 
logical constituent, namely the foot. 
Furthermore, we see that the nuclei are projected 
from the (O R)* line to aiine where they are the 
single elements. If we change the top N's in this 
picture into ~'s we have something like a metrical 
syllable line. 
If we incorporate these two innovations into our 
theory, we can translate th structure in 22 into a 
perfectly normal grid, in fact into a complete linguis- 
tic grid: 
(23) - head-of-foot (-- LTOP, G) (. *) Soot (-) 
(. *) (. *) syllab, const. (~) (*) (*) (.) (*) x-nine(--) 
I I I I m O n e 
Concludingly, we can say that, although KLV's 
syllable representations are somewhat different from 
linguistic grids, two minor adjustments can make 
them isomorphic: 
• in stead of (O R)* we assume (R)(O R)*, i.e. 
there can be onsetless syllables (KLV themselves 
note that most of the (O R)* stipulation can 
be made to follow from independent stipulations 
like interconstituent government). This follows 
a forteriori for the (R)(O R)* stipulation. 
• in stead of 22 we assume VR~ : \[R~ E {~--, }\] ^ R~_,~. =*-- 
The first conjunct of this definition is simply 
my translation of Kager's (\[1989\]) Binary Con- 
stituency Hypothesis 12 and the second con- 
junct does the same as the original definition of 
KLV: it gives the correct choice of government 
for the subsyllabic line. 
As far as I can see, none of these modifications 
alters the empirical scope of KLV's theory in any 
important way. I conclude that for all practical pur- 
poses, KLV's representation of the syllable equals my 
definition of a linguistic grid. 
6 Conclusion 
In this paper we have seen that three more or less 
popular representational systems in modern phonol- 
ogy are notational variants of each other in most 
330 
important ways: these are bracketed grid theory, 
Dependency Phonology and Government Phonology. 
The basic ideas underlying each of these frameworks 
are government/dependency on the one hand and the 
division of a structure into lines on the other. 
The similarity between the frameworks is obscured 
mainly by the immense differences in notation; but 
we have shown that the algebraic systems underlying 
these formalisms is basically the same. 
In \[Maxwell, 1992\] it is shown that the differences 
between Dependency Graphs and X-bar structures as 
used in generative syntax are minimal. It remains to 
be shown whether there are any major formal differ- 
ences between the bracketed grids that are presented 
in this paper and the 'X-bar-structures-cure-lines' as 
they are represented in \[Levin, 1985\] and \[Hermans, 
1990\]. 
Acknowledgements 
I thank Chris Sijtsma, Craig Thiersch and Ben Her- 
marls and the anonymous EACL reviewer of the ab- 
stract for comments and discussion. I alone am re- 
sponsible for all errors. 
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